The Haaretz report said that the talks in Cairo were the first direct meeting between official representatives of the two countries since the fall of the Shah in 1979.
The meetings were held behind closed doors, and all participants committed to complete secrecy to allow a full and frank discussion, the paper said.
But it added that news of the talks was leaked by Australian sources to the Australian daily, The Age.
The reported meetings came amid controversy over Iran's nuclear programme. Western countries, notably the US, accuse Tehran of seeking to develop atomic weapons, but Iran insists its nuclear facilities are for non-military use.
Israel, which neither denies nor confirms possession of nuclear weapons, is said to have any where between 100 and 200 nuclear warheads, making it the sixth-largest nuclear power.
Iran is a signatory to the Nuclear Non-Proliferataion Treaty (NPT), but Israel is not.
Haaretz reported the meetings were held at the Four Seasons Hotel under the auspices of the International Commission on Nuclear Non-Proliferation and Disarmament (ICNND).
They were also attended by representatives of the Arab League, Jordan, Egypt, Tunisia, Turkey, Morocco, the United Arab Emirates and Saudi Arabia, along with European and American officials.
During the meetings, Zafary-Odiz explained the Israeli policy of being willing, in principle, to discuss the Middle East as a nuclear-free zone, according to Haaretz.
The paper added that she detailed Israel's unique strategic situation, saying regional security must be strengthened, security arrangements agreed upon and a peace agreement sealed before Tel Aviv would feel at liberty to discuss nuclear disarmament.
The exchanges between the two officials took place within three panel sessions, the paper reported.
Each session dealt with one of the issues with which the ICNND is concerned - declaring the Middle East a nuclear-free zone, preventing nuclear proliferation in the region and matters of nuclear energy for peaceful purposes.
Zafary-Odiz and Soltanieh did not meet or shake hands outside the sessions.
According to Haaretz, Soltanieh directly asked Zafary-Odiz in one of the sessions: "Do you or do you not have nuclear weapons?"
Zafary-Odiz smiled, but did not respond.